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The COSINE of an arc, is the sine of the complement of that arc, and is equal to the part of the radius comprised between the centre of the circle and the foot of the sine.
The TANGENT of an arc, is a line which touches the circle in one extremity of that arc, and is continued from thence till it meets a' line drawn from or through the centre and through the other extremity of the arc.
The SECANT of an arc, is the radius drawn through one of the extremities of that arc and prolonged till it meets the tangent drawn from the other extremity.
The VERSED SINE of an arc, is that part of the diameter of the circle which lies between the beginning of the arc and the foot of the sine.
The COTANGENT, COSECANT, and coverSED SINE of an arc, are the tangent, secant, and versed sine, of the complement of such arc.
3. Since arcs are proper and adequate measures of plane angles, (the ratio of any two plane angles being constantly equal to the ratio of the two arcs of any circle whose centre is the angular point, and which are intercepted by the lines whose inclinations form the angle), it is usual, and it is perfectly safe, to apply the above names without circumlocution as though they referred to the angles themselves; thus, when we speak of the sine, tangent, or secant, of an angle, we mean the sine, tangent, or secant, of the arc which measures that angle; the radius of the circle employed being known.
4. It has been shown in the 2d vol. (pa. 6), that the tangent is a fourth proportional to the cosine, sine, and radius; the secant, a third proportional to the cosine and radius; the cotangent, a fourth proportional to the sine, cosine, and radius; and the cosecant a third proportional to the sine and radius. Hence, making use of the obvious abbreviations, and converting the analogies into equations, we have
Or, assuming unity for the rad. of the circle, these will become
These preliminaries being borne in mind, the student may pursue his investigations.
5. Let ABC be any plane triangle, of which the side BC opposite the angle a is denoted by the small letter a, the side AC opposite the angle в by the small letter b, and the side AB opposite the angle c by
the small letter c, and CD perpendicular to AB: then is, c = a. cos B + b. cOS A.
For, since AC b, AD is the cosine of a to that radius ; consequently, supposing radius to be unity, we have AD b, COS. A. In like manner it is BD = α . cos. B. Therefore, AD † BD = AB = c = a. cos. B + b. cos. A. By pursuing similar reasoning with respect to the other two sides of the triangle, exactly analogous results will be obtained. Placed together, they will be as below:
6. Now, if from these equations it were required to find expressions for the angles of a plane triangle, when the sides are given; we have only to multiply the first of these equations by a, the second by b, the third by c, and to subtract each of the equations thus obtained from the sum of the other For thus we shall have
b2 + c2
a2 2bc. cos. A, whence cos. A =
7. More convenient expressions than these will be dedu, ced hereafter: but even these will, often be found very convenient, when the sides of triangles are expressed in integers, and tables of sines and tangents, as well as a table of squares, (like that in our first vol.) are at hand.
Suppose, for example, the sides of the triangle are a=320, b=562, c = 800, being the numbers given in prop. 4, pa. 161, of the Introduction to the Mathematical Tables: then we have
The remainder being log. cos. A, or of 18°20'
The remainder being log. cos. B, or of 33°35′ = 9·9207060 Then 180° (18° 20′ + 33° 35′) = 128° 5′ = c; where all the three triangles are determined in 7 lines.
8. If it were wished to get expressions for the sines, instead of the cosines, of the angles; it would merely be necessary to introduce into the preceding equations (marked II),
instead of cos. A, cos. B, &c, their equivalents cos. A=√(1— sin2. A), COS. B =√(1-sin2. B), &c. For then, after a little reduction, there would result,
Or, resolving the expression under the radical into its four constituent factors, substituting s for a+b+c, and reducing, the equations will become
These equations are moderately well suited for computation in their latter form; they are also perfectly symmetrical: and as indeed the quantities under the radical are identical, and are constituted of known terms, they may be represented by the same character; suppose K: then shall we have
Hence we may immediately deduce a very important theorem: for, the first of these equations, divided by the second,
Or, in words, the sides of plane triangles are proportional to the sines of their opposite angles. (See th. 1 Trig. vol. ii).
9. Before the remainder of the theorems, nécessary in the solution of plane triangles, are investigated, the fundamental proposition in the theory of sines, &c, must be deduced, and the method explained by which Tables of these quantities, confined within the limits of the quadrant, are made to extend to the whole circle, or to any number of quadrants whatever. In order to this, expressions must be first obtained for the sines, cosines, &c, of the sums and differences of any two arcs or angles. Now, it has been found (I) that a = b. cos. c + c COS. B. And the equations (IV) give Substituting these va
b = a
sin. c sin. A
lues of b and c for them in the preceding equation, and mul tiplying the whole by it will become
sin. A sin. B. cos. c + sin. c. cos. B.
But, in every plane triangle, the sum of the three angles is two right angles; therefore, B and C are equal to the supplement of A: and, consequently, since an angle and its supplement have the same sine (cor. 1, pa. 3, vol. ii), we have sin. (B+c) = sin. B . cos. c + sin. c. cos. B.
10. If, in the last equation, c become subtractive, then would sin. c manifestly become subtractive also, while the cosine of c would not change its sign, since it would still continue to be estimated on the same radius in the same direction. Hence the preceding equation would become sin, (B-c) sin. B. cos. C sin. c
11. Let c' be the
complement of c, and 4 be the quarter of the circumference: then will c′ = 40-c, sin. c'=cos. c, and cos. c' sin. c. But (art. 10), sin. (B — c') = sin. B. cos. c-sin. c' cos. B. Therefore, substituting for sin. c', cos. c', their values, there will result sin. (Bc') = sin. B . sin. c - Cos. B. cos. C. But because c = 10 c, we have sin. (B-C) sin. (B+C-40) = sin. [(B+c)−÷O] = sin. [O (B+c)]= cos. (B+c). Substituting this value of sin. (B-c') in the equation above, it becomes cos. (B+c) = COS. B. COS. C sin. B. sin. c.
12. In this latter equation, if c be made subtractive, sin. c will become sin. c, while cos. c will not change: consequently the equation will be transformed to the following, viz, cos. (B-c) = COS. B cos. c + sin. B. sin. c.
If, instead of the angles B and C, the angles had been a and B; or, if A and B represented the arcs which measure those angles, the results would evidently be similar they may therefore be expressed generally by the two following equations, for the sines and cosines of the sums or differences of any two arcs or angles:
13. We are now in a state to trace completely the mutations of the sines, cosines, &c, as they relate to arcs in the various parts of a circle; and thence to perceive that tables which apparently are included within a quadrant, are, in fact, applicable to the whole circle.
Imagine that the radius MC of the circle, in the marginal figure, coinciding at first with AC, turns about the point c (in the same manner as a rod would turn on a pivot), and thus
forming successively with AC all possible angles: the point м at its extremity passing over all the points of the circumference ABA'B'A, or describing the whole circle. Tracing this motion attentively, it will appear, that at the point A, where the arc is nothing, the sine is nothing also, while the cosine does not differ from the radius. As the radius MC recedes from AC, the sine PM keeps increasing, and the cosine CP decreasing, till the describing point м has passed over a quadrant, and arrived at B in that case, PM becomes equal to CB the radius, and the cosine CP vanishes. The point м continuing its motion beyond B, the sine P'M' will diminish, while the cosine cr', which now falls on the contrary side of the centre c will increase. In the figure, P'M' and CP' are respectively the sine and cosine of the arc a'm', or the sine and cosine of ABM', which is the supplement of A'M' to O, half the circumference: whence it follows that an obtuse angle (measured by an arc greater than a quadrant) has the same sine and cosine as its supplement; the cosine however, being reckoned subtractive or negative, because it is situated contrariwise with regard to the centre c.
When the describing point м has passed over O, or half the circumference, and has arrived at a', the sine P'M' vanishes, or becomes nothing, as at the point A, and the cosine is again equal to the radius of the circle. Here the angle ACM has attained its maximum limit; but the radius CM may still be supposed to continue its motion, and pass below the diameter AA'. The sine, which will then be "M", will consequently fall below the diameter, and will augment as M moves along the third quadrant, while on the contrary CP", the cosine, will diminish. In this quadrant too, both sine and cosine must be considered as negative; the former being on a contrary side of the diameter, the latter a contrary side of the centre, to what each was respectively in the first quadrant. At the point B', where the arc is three-fourths of the circumference, or O, the sine P"M" becomes equal to the radius CB, and the cosine CP" vanishes. Finally, in the fourth quadrant, from B' to A, the sine P"M"", always below AA', diminishes in its progress, while the cosine cr", which is then found on the same side of the centre as it was in the first quadrant, augments till it becomes equal to the radius CA. Hence, the sine in this quadrant is to be considered as pega